**Topics in PDE**

We read selected chapters from classical textbooks on PDE such as:

- Brezis:
*Functional Analyis, Sobolev Spaces and Partial Differential Equations* - Evans:
*Partial Differential Equations* - Gilbarg/Trudinger:
*Elliptic Partial Differential Equations of Second Order* - Krylov:
*Lectures on Elliptic and Parabolic Equations in Hölder Spaces*

The plan is to prepare the reading material in advance and then discuss the material and exercises during the meetings.

In this Cluster Group, the fundamental principles of Kolmogorov and Fokker-Planck equations are explored, which are indispensable tools for comprehending stochastic processes and the dynamics of continuous systems influenced by randomness. The seminar encompasses their theoretical foundations and applications.

Key topics to be covered include:

- Review of Stochastic Processes
- Understanding Diffusion Processes
- Stochastic Differential Equations (SDE)
- Chapman-Kolmogorov Equation
- Kolmogorov Equation
- Fokker-Planck Equation
- Discussion on the Fractional Laplace Operator within a Fokker-Planck type equation.

Initially, we will delve into the classical theory, bridging the gap between stochastic processes and Partial Differential Equations (PDEs). Subsequently, we aim to explore a neoclassical approach, including the use of the fractional Laplacian as an illustrative example.

- Boundary Hölder regularity of solutions to the fractional Laplace equation on domains
- Interior regularity for solutions to the fractional Laplace equation via Riesz-potentials and classical theory
- Barriers and maximum principles
- Optimal boundary regularity
- Higher order regularity by renormalising using powers of the distance function
- Upper Green function bounds for the radially symmetric $\alpha$-stable Levy-process on a ball and on $C^{1,1}$-domains
- Deriving upper Poisson-kernel estimates from Green function estimates

Based on Ros-Oton–Serra's *The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary*, Silvestre's *Regularity of the obstacle problem for a fractional power of the laplace operator* and Chen–Song's *Estimates on Green function and Poisson kernels for symmetric stable processes*.

- Divergence Theorem, Laplace Equation and properties of harmonic functions.
- Heat Equation.
- Sobolev spaces, Compact embeddings, Extensions and Sobolev Inequalities.
- Maximal Function, weak type (1,1), and strong (p,p) estimates.
- Distributions: Regular and irregular distributions, tempered distributions and the Fourier Transform, Convultion, Schwartz Kernel Theorem.

Reading sources included Evan's *Partial Differential Equations*, Brezis' *Functional Analysis, Sobolev Spaces, and Partial Differential Equations*, Stein's *Singular Integrals and Differentiability Properties of Functions* and Duistermaat and Kolk's *Distributions*.

**Reading** *Introduction to Riemannian manifolds* by John M. Lee, supplemented by *Heat kernel and analysis on manifolds* by Alexander Grigor'yan.

- Review of Smooth Manifolds
- Tensor Bundles and Tensor Fields
- Differentials, Differenial Forms and Exterior Derivatives
- Riemannian Metrics on Manifolds
- Construction of Riemannian Metrics on Submanifolds
- Riemannian Volume Form
- Detailed Analysis of the Laplace–Beltrami Operator

- Motivation for Interacting Particle Systems coming from physics
- Background and basic notation on Interacting Particle Systems, basics of mean field models
- Time continuous Markov chains and holding times
- Basic techniques like Coupling, Duality and Monotonicity based on Chapter II of the Book 'interaction particle systems' by Thomas Liggett
- Connection between transition probabilities and transition rates
- Detailed analysis of spin systems and contact processes

- different types of dynamical systems: Topological dynamics, Smooth dynamics, Ergodicity, Hyperbolic dynamics
- discuss examples
- connection between hyperbolic dynamics and differential equations
- discuss about uniform hyperbolicity and chaos
- generalization of uniformly hyperbolic to partially hyperbolic dynamical systems
- discuss on hyperbolic dynamics with singularities and examples including the Lorenz system of ODE

- Deep Learning Architectures (Ovidiu Calin)
- Mathematics for Machine Learning (Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon Ong)

- defining differential operator (as gradient and divergence) on manifold
- understand how Laplacian is defined thanks to Laplace-Beltrami operator
- understand PDEs on manifold and define the Brownian motion on it
- how to construct the Brownian motion on Riemanian manifold with the Laplace-Beltrami operator
- several concrete examples

- Newtonian Classical Mechanics (affine spaces, Galilean transformations, Newton's equation)
- Newtonian Classical Mechanics (examples, kinetic and potential energy)
- Lagrangian Classical Mechanics (motivation, calculus of variations)
- Lagrangian Classical Mechanics (Lagrangian function and action, Euler-Lagrange formula, Hamilton's equation)
- Observables (algebra of observables, Poisson-bracket, analogies to quantum mechanics)
- Axioms of Quantum Mechanics (states, observables, measurement, correspondence principle, position and momentum operators)
- Schrödinger Equation (wavefunction, Hamilton operator, eigenvalue equation and eigenstates)
- Schrödinger Equation continued (eigenstates for potential well and potential step), Double-Slit Experiment (interference, probability amplitude calculus)

- Tangent Vectors, The Differential of a Smooth Map, The Tangent Bundle (reference: “Introduction to Smooth Manifolds” by John M. Lee)
- Connection, Vector Fields, Covariant Derivative, Geodesic and Parallel Translation
- Discussion about curvature of space and Gauss-Bonnet Theorem
- Divergence theorem, Green's identities
- Laplace–Beltrami operator
- Lie group and principal bundle

- differentiable structures, manifolds, tangent spaces
- de Rham cohomology, embeddings of manifolds, regular value theorem
- (pseudo-)Riemannian manifolds, Riemannian measures, Laplace-Beltrami operator, Gauss-Green identity, Stokes' theorem
- Covariant derivative, Riemannian connection, Christoffel symbols, Parallel displacement, exponential mapping (cf. Section 5D, Kuehnel: Differential geometry)
- Curvature tensor and intrinsic geometry of surfaces: Equations of Gauss and Weingarten, Theorema egregium, Fundamental theorem of the local theory of surfaces (Sections 4A-D, Kuehnel)
- More on the curvature tensor and Weingarten map, Sectional curvature, Ricci tensor and curvature, Einstein tensor (Chapters 4,6: Kuehnel)
- Spaces of constant curvature: Hyperbolic space, Geodesics and Jacobi fields, Local isometry of spaces of constant curvature (Chapter 7: Kuehnel)

- Organizational meeting
- Probability that random polynomial has no real roots (Talk by Anna Gusakova)
- Theorems on positive real functions (Talk by Anna Muranova)
- Non-explosion of solutions to SDEs with singular coefficients (Talk by Chengcheng Ling)
- Emergence of Flocking in the Fractional Cucker-Smale Model (Talk by Peter Kuchling)
- Robust Poincaré inequality and Application to the transitional phase of fractional Laplacian (Talk by Guy Fabrice Foghem Gounoue)
- My problems and failed ideas regarding the edge fluctuation of non hermitian random matrices with independent entries (Jonas Jalowy)
- Effective impedance of a finite electric network. Definitions and main properties (theorems and conjectures) (Anna Muranova)
- Application of Probability Methods in Number Theory and Integral Geometry (Anna Gusakova)
- Ordered fields. The ordered field of rational functions“ (Anna Muranova)
- Dynamics on the cone: an overview of my thesis (Peter Kuchling)
- Well-posedness of SDE driven by Levy noise (Chengcheng Ling)
- Discussion about the talks for workshop/retreat

- Point processes 3: Moment Problem, Local Convergence Jansen, Gibbsian Point Processes, pp. 33 - 45
- Point processes 2: Intensity Measure, Correlation Functions, Generating Functionals Jansen, Gibbsian Point Processes, pp. 25 - 33
- Point processes 1: Configuration Spaces, Observables, PPP Jansen, Gibbsian Point Processes, pp. 15 - 25
- Introduction to partition functions
- Partition Functions and Gibbs Measures
- Correlation Functions
- Phase Transitions (Critical Exponents and Models)
- Scaling Limits
- Mean Field Limit of the Cucker-Smale model: Two Approaches (Peter Kuchling)

**Reading** *Spectral graph theory* by Fan R. K. Chung

- Chapter 2: Isoperimetric problems
- Chapter 3: Diameters and eigenvalues
- Talk by Anna Muranova: On effective resistance of electric network with impedances
- Talk by Melissa Meinert: Ollivier Ricci curvature for general Laplacians
- Markov chains
- Talk by Filip Bosnic: Examples of Poincaré and log-Sobolev inequalities on discrete configuration spaces
- Introduction to random graphs by Alan Frieze and Michał Karoński

**Discuss** *Lévy Processes and Infinitely Divisible Distributions* by Ken-iti Sato

- Chapter 1 (pp. 1 - 30): Examples of Levy Processes

- Chapter 2 (pp. 31 - 68): Characterisation and Existence of Levy Processes and Additive Processes
- Chapter 2 (pp. 31 - 41): Infinitely Divisible Processes and the Lévy-Khintchine formula
- Chapter 2 (pp. 54 - 67): Transition Functions and the Markov Property; Existence of Lévy and Additive Processes
- Chapter 3 (pp. 69 - 77): Selfsimilar and Semi-selfsimilar Processes and their Exponents
- Chapter 3 (pp. 77 - 90): Representations of Stable and Semi-stable Distributions
- Viswanathan et al. Papers: “Levy Flight Search Patterns of Wandering Albatrosses” and “Optimizing the Success of Random Searches”
- Bertoin (pp. 18-24): Potential theory: Markov property and important definitions
- Bertoin (pp. 43-48): Duality and time reversal
- Bertoin (pp. 48-56): Potential theory: capacity measure; essentially polar sets and capacity
- Bertoin (pp. 56-68): Potential theory: energy; the case of a single point
- Bertoin (pp. 71-84): Subordinators: definitions; passage across a level; arcsine laws
- Bertoin (pp. 84-99): Subordinators: rates of growth; (Hausdorff-)dimension of the range

**Discuss**

- Maximum principle for ecliptic operators (
*Gibarg-Trudinger*) - Paley-Littlewood Decomposition (
*Grafakos classical Fourrier Analysis, second or third edtion*) - Another looks at Sobolev spaces (
*Articles Brezis Bourgain_Mironescu*) - Spectral theory: spectral measure for Unbounded operators
- Spectral theory for unbounded operator and spectral measure (Book by Reed-Simon)
- Restriction of the Fourier Transform on the sphere (Book by Elias Stein)
- Interplation of Banach spaces and application (Lecture note by Alessandra Lunadri Theory of interpolation 2009)
- Introduction to optimal transport theory (Book by Cedri Villani: Old and New, it is free on-line)
- Introduction to control and non linearity (Book by Jean-Michel Coron: Control and Non-linearity free one his we-page)
- Introduction to Calculus of variations